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Corrigendum to “Interpolatory estimates in monotone piecewise polynomial approximation” [J. Approx. Theory 223 (2017) 1–8]
Available online at www.sciencedirect.com
ScienceDirect Journal of Approximation Theory 228 (2018) 79–80 www.elsevier.com/locate/jat
Corrigendum
Corrigendum to “Interpolatory estimates in monotone piecewise polynomial approximation” [J. Approx. Theory 223 (2017) 1–8] D. Leviatan a , ∗, 1 , I.L. Petrova b a Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel b Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
Received 10 December 2017; received in revised form 13 December 2017; accepted 19 December 2017 Available online 12 January 2018 Communicated by Paul Nevai
Abstract We correct a couple of misprints in the above mentioned article. c 2017 Elsevier Inc. All rights reserved. ⃝ MSC: 41A10; 41A25; 41A29; 41A30 Keywords: Monotone approximation by piecewise polynomials; Degree of pointwise approximation; Jackson-type interpolatory estimates
In the last displayed formula in the proof of [1, Lemma 3.1], there is an inadvertent misprint. The correct formula is: For 0 < x ≤ h ≤ H , ′ L r,h (x) =
r ∑ f (i) (0) i−1 x + ar (h; f )(r + 1)x r (i − 1)! i=k
DOI of original article: http://dx.doi.org/10.1016/j.jat.2017.07.006.
∗ Corresponding author.
E-mail addresses: [email protected] (D. Leviatan), [email protected] (I.L. Petrova). 1 Part of this work was done while the author was visiting Taras Shevchenko National University of Kyiv.
https://doi.org/10.1016/j.jat.2017.12.003 c 2017 Elsevier Inc. All rights reserved. 0021-9045/⃝
80
D. Leviatan, I.L. Petrova / Journal of Approximation Theory 228 (2018) 79–80
( r ) ∑ f (i) (0) = x k−1 x i−k + ar (h; f )(r + 1)x r −k+1 (i − 1)! ) ( i=k r ∑ f (k) (0) | f (i) (0)| i−k (r + 1)ω1 (H ) r −k k−1 − H − H ≥x (k − 1)! i=k+1 (i − 1)! r! > 0. The reader should observe that in the line before the last line ω1 (1) has been replaced by ω1 (H ), and the last exponent is r − k instead of r − k + 1. References [1] D. Leviatan, I.L. Petrova, Interpolatory estimates in monotone piecewise polynomial approximation, J. Approx. Theory 223 (2017) 1–8.
ScienceDirect Journal of Approximation Theory 228 (2018) 79–80 www.elsevier.com/locate/jat
Corrigendum
Corrigendum to “Interpolatory estimates in monotone piecewise polynomial approximation” [J. Approx. Theory 223 (2017) 1–8] D. Leviatan a , ∗, 1 , I.L. Petrova b a Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel b Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
Received 10 December 2017; received in revised form 13 December 2017; accepted 19 December 2017 Available online 12 January 2018 Communicated by Paul Nevai
Abstract We correct a couple of misprints in the above mentioned article. c 2017 Elsevier Inc. All rights reserved. ⃝ MSC: 41A10; 41A25; 41A29; 41A30 Keywords: Monotone approximation by piecewise polynomials; Degree of pointwise approximation; Jackson-type interpolatory estimates
In the last displayed formula in the proof of [1, Lemma 3.1], there is an inadvertent misprint. The correct formula is: For 0 < x ≤ h ≤ H , ′ L r,h (x) =
r ∑ f (i) (0) i−1 x + ar (h; f )(r + 1)x r (i − 1)! i=k
DOI of original article: http://dx.doi.org/10.1016/j.jat.2017.07.006.
∗ Corresponding author.
E-mail addresses: [email protected] (D. Leviatan), [email protected] (I.L. Petrova). 1 Part of this work was done while the author was visiting Taras Shevchenko National University of Kyiv.
https://doi.org/10.1016/j.jat.2017.12.003 c 2017 Elsevier Inc. All rights reserved. 0021-9045/⃝
80
D. Leviatan, I.L. Petrova / Journal of Approximation Theory 228 (2018) 79–80
( r ) ∑ f (i) (0) = x k−1 x i−k + ar (h; f )(r + 1)x r −k+1 (i − 1)! ) ( i=k r ∑ f (k) (0) | f (i) (0)| i−k (r + 1)ω1 (H ) r −k k−1 − H − H ≥x (k − 1)! i=k+1 (i − 1)! r! > 0. The reader should observe that in the line before the last line ω1 (1) has been replaced by ω1 (H ), and the last exponent is r − k instead of r − k + 1. References [1] D. Leviatan, I.L. Petrova, Interpolatory estimates in monotone piecewise polynomial approximation, J. Approx. Theory 223 (2017) 1–8.